How can I make a curve higher resolution?
You can increase the 'Resolution' of the curve from the 'Data' tab while the curve is selected:



Bare in mind, the higher you set this, the more geometry that has to render, so renders may become slower.

You only need alter the 'Preview' resolution if you like, when the 'Render' resolution is 0 it will default to whatever value the 'Preview' is set at. Alternatively if the viewport was becoming too slow you could have a lower 'Preview' resolution and 'Render' set higher.

If you have given the curve a bevel you can increase the resolution of the bevel by increasing the 'Resolution' value in the 'Geometry' panel in the curve properties.